All of the 4th grade teachers and students from Springer went on a field trip to an archaeology museum. Tickets were $$5.00$ each for teachers and $$3.50$ each for students, and the group paid $$41.50$ in total. The next month, the same group visited a natural history museum where the tickets cost $$10.00$ each for teachers and $$9.50$ each for students, and the group paid $$105.50$ in total. Find the number of teachers and students on the field trips.
Explanation: Let $x$ equal the number of teachers and $y$ equal the number of students. The system of equations is: ${5x+3.5y = 41.5}$ ${10x+9.5y = 105.5}$ Solve for $x$ and $y$ using elimination. Multiply the top equation by $-2$ ${-10x-7y = -83}$ ${10x+9.5y = 105.5}$ Add the top and bottom equations together. $ 2.5y = 22.5 $ $ y = \dfrac{22.5}{2.5}$ ${y = 9}$ Now that you know ${y = 9}$ , plug it back into $ {5x+3.5y = 41.5}$ to find $x$ ${5x + 3.5}{(9)}{= 41.5}$ $5x+31.5 = 41.5$ $5x = 10$ $x = \dfrac{10}{5}$ ${x = 2}$ You can also plug ${y = 9}$ into $ {10x+9.5y = 105.5}$ and get the same answer for $x$ ${10x + 9.5}{(9)}{= 105.5}$ ${x = 2}$ There were $2$ teachers and $9$ students on the field trips.